Bases for Irreducible Representations of U2n in the Chain U2n⊃Un+̇Un

Abstract

The highest weight polynomials of irreducible representations of a U₃ × U₃ subgroup of U₆, which occur in the reduction of a given irreducible representation of U₆, are obtained, making use of Littlewood's rules for the reduction of an irreducible representation of U₆ with respect to a U₃ × U₃ subgroup. The various irreducible representations of U₃ × U₃ occurring in the reduction are uniquely labeled by parameters which are obtained in a natural and obvious way from Littlewood diagrams. These polynomials could not be obtained by the algebraic methods developed so far. The method described here overcomes the division problem encountered there and makes the solution possible. It can also be directly extended to the case of $U_{\text{2n}}{\mathrm{\supset U}}_n{\mathrm{+̇U}}_n$ for any n.